Analytical solutions of PDEs by unique polynomials for peristaltic flow of heated Rabinowitsch fluid through an elliptic duct

In this research, we have considered the convective heat transfer analysis on peristaltic flow of Rabinowitsch fluid through an elliptical cross section duct. The Pseudoplastic and Dilatant characteristics of non-Newtonian fluid flow are analyzed in detail. The Rabinowitsch fluid model shows Pseudoplastic fluid nature for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma > 0$$\end{document}σ>0 and Dilatant fluid behaviour for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma < 0.$$\end{document}σ<0. The governing equations are transformed to dimensionless form after substituting pertinent parameters and by applying the long wavelength approximation. The non-dimensional momentum and energy equations are solved analytically to obtain the exact velocity and exact temperature solutions of the flow. A novel polynomial of order six having ten constants is introduced first time in this study to solve the energy equation exactly for Rabinowitsch fluid flow through an elliptic domain. The analytically acquired solutions are studied graphically for the effective analysis of the flow. The flow is found to diminish quickly in the surrounding conduit boundary for Dilatant fluid as compared to the Pseudoplastic fluid. The temperature depicted the opposite nature for Pseudoplastic and Dilatant fluids. The flow is examined to plot the streamlines for both Pseudoplastic and Dilatant fluids by rising the flow rate.

The process in which fluid flows across a conduit by the sinusoidal fluctuating boundaries is known as Peristalsis. The fluid propels by the means of deformation of the walls parallel to the axis of channel in peristaltic flow. The wide applications of peristaltic flow in the industrial area, physiological and engineering fields like corrosive fluid transport, blood pumps in heart lungs machines, blood flow in small vessels makes it much important. Due to these many applications numerous researchers has been studying the peristaltic flow via various conduits. Bohm and Friedrich studied the flow of viscoelastic fluids under peristaltic wall deformation of channel by considering an assumption of small Reynold number 1 16 gave the analysis of solid particles influence on the peristaltically flowing Jeffery fluid across eccentric annuli. Vaidya et al. 17,18 studied the influence of variable liquid and rheological characteristics on peristaltic flow of Rabinowitsch fluid across an inclined channel. The variables properties and application of peristaltic flow of non-Newtonian Rabinowitsch fluid under various effects is discussed by many researchers [19][20][21][22] .
The heat transfer effects on fluid flowing through different channels are studied by many researchers. The peristaltic flow of heated magnetohydrodynamic fluid under partial slip influence is analyzed by Nadeem and Akram 23 . Akbar and Butt 24 provided the analytical study of nanofluid flow in curved tube with vibrating walls. Bibi and Xu 25 presented the heat transfer in a nanofluid flow through a horizontal conduit with MHD effects. Raza et al. 26 gave the study of water-based nanofluid for various shaped nanoparticles in an asymmetric conduit having permeable boundary under magnetic effects. The influence of heat flux and endoscope on non-Newtonian Jeffery fluid movement across concentric symmetric tubes due to peristaltic motion of walls is discussed by Abd-Alla and Abo-Dahab 27 . Abbasi et al. 28 examined the nanofluids flow under the peristaltic and electroosmotic effects via asymmetric narrow channel. Li et al. 29 studied the peristaltic movement of non-Newtonian nanofluid by utilizing Jeffery fluid model under the hall effects and viscous dissipation. Some of the recent literature work that highlights the peristaltic flow, pseudo plastic flow characteristics, MHD impacts and chemical reactions on a channel flow problem are given as [30][31][32][33] .
In this research, we have studied the heated Rabinowitsch fluid flow across a duct of elliptical cross section. The published literature and researchers clearly highlight that there is no research on Peristaltic flow of heated Rabinowitsch fluid inside an elliptic duct. The non-Newtonian fluid flow is studied through a channel with sinusoidally fluctuating boundary walls. The velocity and heat transfer effects on fluid flow are analyzed. A unique polynomial is involved to obtain the solution of temperature. The dilatant and pseudoplastic characteristics of the Rabinowitsch fluid are examined.

Mathematical formulation
Consider the flow of a heated Rabinowitsch fluid through a duct having elliptical cross section. The elliptic shape conduit is studied with sinusoidally fluctuating walls. The mathematical study in completed by utilizing cartesian coordinates (X, Y , Z) . The sinusoidally deformed walls are taken as 34   where, σ is coefficient of pseudoplasticity. The Rabinowitsch fluid shows Pseudoplastic, Newtonian, Dilatant behaviour for σ > 0, σ = 0, σ < 0 respectively. The transformations relate the fixed (unsteady) and wave (steady) frames are given following The dimensionless variables to transform the equations into non-dimensional form where, D h is the hydraulic diameter of the ellipse and e is known as the eccentricity of the ellipse such that 0 < e < 1 and e = √ By applying long wavelength approximation together with the transformation given in (9) and non-dimensional variables by ignoring dash notation provided in (10)  1 − e 2 Sin 2 (α)dα.

Results and discussion
The graphical analysis of analytically attained results in earlier fragment is presented in this section. This analysis provides an effective study and better understanding of the physical aspects of peristaltic motion of Rabinowitsch fluid flow across elliptic duct. The effect of different physical constraints on the flow velocity, temperature distribution, pressure gradient and pressure rise are discussed. The three-dimensional graphs of velocity and temperature distribution are provided in this graphical examination.

Conclusions
In this study, we have considered the flow of heated Rabinowitsch fluid across an elliptical cross section duct. The flow is studied with sinusoidally vibrating walls of the conduit. The impact of physical constraints on Pseudoplastic and Dilatant fluids through elliptic duct is analyzed. A novel mathematical technique is introduced to solve the partial differential equation that includes the heat transfer in present analysis. A polynomial of order six with ten constants is introduced first time in such analysis to solve the temperature equation. The major outcomes of our study are following: The velocity of flow has same magnitude on the axis of conduit. The flow velocity decreases rapidly in neighboring of walls for Dilatant fluid than Pseudoplastic fluid. The temperature has opposite behavior for Pseudoplastic and Dilatant fluids. Our major task was to solve the complex partial differential equation involved in convection heat transfer and we have provided an exact solution for temperature equation by considering a polynomial of order six with ten constants. Such a polynomial is considered first time in this work to solve the energy equation exactly. The pressure gradient shows the periodic behavior. The pressure rise nature of Dilatant fluid opposes the behavior of pressure rise of Pseudoplastic fluid. The velocity and temperature profiles are parabolic in nature for Pseudoplastic fluid. The streamlines analysis unfolds that size of trappings reduces with rise in flow rate for Pseudoplastic fluid but has opposite behavior for Dilatant fluid.